Network evolution with mesoscopic delay

TL;DR

This paper develops probabilistic models to analyze how mesoscopic delays affect the evolution of networks, focusing on local and global properties using stochastic approximation techniques.

Contribution

It introduces new probabilistic tools to understand the asymptotic behavior of networks with mesoscopic delays, extending prior models to include time-lagged information flow.

Findings

Analysis of local neighborhood evolution

Insights into degree distribution dynamics

Understanding of initial node degree growth

Abstract

Owing to the influence of real-world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber-security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models incorporating network delay, where new participants receive information from a time-lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper explores the regime of macroscopic delays in the evolution of the network.